Article ID: MTJPAM-D-23-00046

Title: A symmetric expansion of an analytical function in terms of quadratic functions involving the fractional derivative and some applications to special functions


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-23-00046; Volume 6 / Issue 3 / Year 2024, Pages 83-106

Document Type: Research Paper

Author(s): ‪Richard Tremblay a

aDépartement d’Informatique et Mathématique, Université du Québec à Chicoutimi, Chicoutimi, Qué., Canada G7H 2B1

Received: 18 November 2023, Accepted: 27 November 2023, Published: 5 February 2024

Corresponding Author: Richard Tremblay (Email address: rtrembla@uqac.ca)

Full Text: PDF

Abstract

Motivated by the generalization of Taylor’s series of f(z) obtained by Osler (Taylor’s series generalized for fractional derivatives and applications, SIAM J. Math. Anal. 2, 37–48, 1971), Tremblay and Fugère presented in 2007 a new expansion of an analytic function f(z) in R in terms of a power series θ(t)=tq(t), where q(t) is any regular function and t is equal to the quadratic function [(z − z1)(z − z2)], z1 ≠ z2, where z1 and z2 are two points in R (The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions, Appl. Math. Comput. 187 (1), 507–529, 2007). To illustrate the concept, if q(t)=1, the coefficient of [(z − z1)(z − z2)]n in the power series of the function (z − z1)α(z − z2)βf(z) is

In this paper, using the transformation formula for fractional derivatives obtained by Tremblay, Gaboury and Fugère (A new transformation formula for fractional derivatives with applications, Integral Transforms Spec. Funct. 24 (3), 172–186, 2013)

we present a new form of expansion in terms of quadratic functions symmetric with respect to parameters α and β as well as applications to special functions. Finally, several definite integers of Fourier type are evaluated, in particular an integral due to Ramanujan is generalized.

Keywords: Fractional derivatives, Taylor’s theorem, power series, special functions, Ramanujan integral, Fourier’s integral

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Cite this article

How to cite this article: ‪‪R. Tremblay, A symmetric expansion of an analytical function in terms of quadratic functions involving the fractional derivative and some applications to special functions, Montes Taurus J. Pure Appl. Math. 6 (3), 83-106, 2024; Article ID: MTJPAM-D-23-00046.